Geometric quantization via SYZ transforms

TL;DR

This paper uses SYZ transforms to demonstrate that in certain geometric settings, the space of wave functions remains consistent across different polarizations, resolving a key issue in geometric quantization.

Contribution

It introduces a novel application of SYZ transforms to establish polarization independence in geometric quantization for specific manifolds.

Findings

Wave function spaces are canonically isomorphic across polarizations

SYZ transforms provide a bridge between real and complex polarizations

Results apply to semi-flat Lagrangian torus fibrations and projective toric manifolds

Abstract

The so-called quantization problem in geometric quantization is asking whether the space of wave functions is independent of the choice of polarization. In this paper, we apply SYZ transforms to solve the quantization problem in two cases: (1) semi-flat Lagrangian torus fibrations over complete compact integral affine manifolds, and (2) projective toric manifolds. More precisely, we prove that the space of wave functions associated to the real polarization is canonically isomorphic to that associated to a complex polarization via SYZ transforms in both cases.